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Chapter 2: Problem 24

Perform the addition or subtraction and write the result in standard form. $$22+(-5+8 i)-9 i$$

Short Answer

Expert verified
The result of the addition and subtraction in standard form is \(17 - i\).

Step by step solution

01

Identify Real and Imaginary Parts

First, identify the real and imaginary parts of each number. For \(22\), the real part is 22 and there is no imaginary part. For \(-5+8i\), the real part is -5 and the imaginary part is \(8i\). For \(-9i\), there is no real part and the imaginary part is \(-9i\).
02

Sum Real Parts and Imaginary Parts Separately

Next, add the real parts together and add the imaginary parts together. The sum of the real parts is \(22 - 5 = 17\). The sum of the imaginary parts is \(8i - 9i = -1i\).
03

Write the Result in Standard Form

Finally, write the result in standard form. In this case, our result is \(17 - 1i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of Complex Numbers
The standard form of a complex number is expressed as a + bi, where a represents the real part, b is the coefficient of the imaginary part, and i is the imaginary unit, satisfying the equation i^2 = -1. The real part can be any real number, while the imaginary part multiplies the unit i to indicate that it is on a different axis than the real numbers when graphed on the complex plane.

For example, the number -5 + 8i has a real part of -5 and an imaginary coefficient of 8, which is in standard form. Standard form is useful because it clearly differentiates between the two types of components in a complex number, allowing for straightforward implementation of algebraic operations.
Imaginary Parts of Complex Numbers
The imaginary part of a complex number is the term that includes the imaginary unit i. It represents a dimension perpendicular to the real number line. For the complex number -5 + 8i, the 8i is the imaginary part. When performing operations with complex numbers, the imaginary parts are treated separately from the real parts.

Imaginary parts can be added or subtracted from one another just like real numbers. When adding complex numbers, you simply combine the imaginary parts algebraically. For instance, 8i - 9i equals -1i.
Real Parts of Complex Numbers
The real part of a complex number is the term that is not multiplied by the imaginary unit i. It is the component that lies on the horizontal axis of the complex plane, which is identical to the real number line. In the complex number 22 + (-5 + 8i) - 9i, 22 and -5 are the real parts.

Real parts can be combined independently of imaginary parts. They follow the same rules as adding or subtracting real numbers. In the given example, combining the real parts 22 and -5 gives us 17.
Algebraic Operations with Complex Numbers
Algebraic operations with complex numbers follow the standard rules of algebra, adapted for the presence of an imaginary unit. Addition and subtraction involve combining real parts with real parts and imaginary parts with imaginary parts separately. Multiplication and division require the use of the distributive property and the fact that i^2 = -1.

For the example 22 + (-5 + 8i) - 9i, the real parts 22 and -5 are added to give 17 while the imaginary parts 8i and -9i are combined to yield -1i. Consequently, the result in standard form is 17 - 1i, following the algebraic operations for complex numbers.

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Most popular questions from this chapter

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x^{4}-29 x^{2}+100$$

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